6-1 Additional Practice Answer Key

6-1 Additional Practice: Mastering the Fundamentals (Answer Key & Detailed Explanations)

This comprehensive guide provides the answer key and detailed explanations for a hypothetical "6-1 Additional Practice" assignment, commonly found in mathematics or science textbooks at a middle school or early high school level. While a specific textbook isn't referenced, the problems and solutions below cover a range of fundamental concepts frequently encountered in these grades. This resource aims to not only provide the correct answers but also to thoroughly explain the underlying principles, boosting your understanding and problem-solving skills. We'll cover various topics, focusing on clear explanations and strategies to approach similar problems in the future.

Section 1: Algebra Basics (Problems 1-5)

Problem 1: Solve for x: 3x + 7 = 16

Answer: x = 3

Explanation: To solve for x, we need to isolate it on one side of the equation. First, subtract 7 from both sides: 3x = 9. Then, divide both sides by 3: x = 3.

Problem 2: Simplify the expression: 2(4a - 5b) + 3(a + 2b)

Answer: 11a - 4b

Explanation: Distribute the 2 and 3 to the terms inside the parentheses: 8a - 10b + 3a + 6b. Combine like terms: (8a + 3a) + (-10b + 6b) = 11a - 4b.

Problem 3: Solve the inequality: 5x - 2 > 13

Answer: x > 3

Explanation: Add 2 to both sides: 5x > 15. Then, divide both sides by 5: x > 3. Remember that when you multiply or divide an inequality by a negative number, you must reverse the inequality sign.

Problem 4: Find the slope of the line passing through points (2, 5) and (4, 9).

Answer: m = 2

Explanation: The slope (m) of a line passing through points (x1, y1) and (x2, y2) is given by the formula: m = (y2 - y1) / (x2 - x1). Plugging in the given points: m = (9 - 5) / (4 - 2) = 4 / 2 = 2.

Problem 5: Solve the system of equations:

x + y = 7 x - y = 1

Answer: x = 4, y = 3

Explanation: You can solve this system using either substitution or elimination. Using elimination, add the two equations together: (x + y) + (x - y) = 7 + 1. This simplifies to 2x = 8, so x = 4. Substitute x = 4 into either equation (let's use the first one): 4 + y = 7, which gives y = 3.

Section 2: Geometry Fundamentals (Problems 6-10)

Problem 6: Find the area of a rectangle with length 8 cm and width 5 cm.

Answer: 40 cm²

Explanation: The area of a rectangle is calculated by multiplying its length and width: Area = length × width = 8 cm × 5 cm = 40 cm².

Problem 7: Find the circumference of a circle with a radius of 7 inches. (Use π ≈ 22/7)

Answer: 44 inches

Explanation: The circumference of a circle is given by the formula: Circumference = 2πr, where r is the radius. Using π ≈ 22/7 and r = 7 inches: Circumference = 2 × (22/7) × 7 inches = 44 inches.

Problem 8: Calculate the volume of a cube with side length 4 meters.

Answer: 64 m³

Explanation: The volume of a cube is calculated by cubing its side length: Volume = side³ = 4 m × 4 m × 4 m = 64 m³.

Problem 9: Find the area of a triangle with base 10 cm and height 6 cm.

Answer: 30 cm²

Explanation: The area of a triangle is given by the formula: Area = (1/2) × base × height = (1/2) × 10 cm × 6 cm = 30 cm².

Problem 10: What is the measure of each interior angle of a regular hexagon?

Answer: 120°

Explanation: The sum of the interior angles of an n-sided polygon is given by the formula: (n - 2) × 180°. For a hexagon (n = 6), the sum is (6 - 2) × 180° = 720°. Since it's a regular hexagon, all angles are equal, so each angle measures 720° / 6 = 120°.

Section 3: Number Sense and Operations (Problems 11-15)

Problem 11: What is the greatest common factor (GCF) of 18 and 24?

Answer: 6

Explanation: Find the prime factorization of each number: 18 = 2 × 3 × 3 and 24 = 2 × 2 × 2 × 3. The GCF is the product of the common prime factors raised to the lowest power: 2 × 3 = 6.

Problem 12: What is the least common multiple (LCM) of 12 and 18?

Answer: 36

Explanation: Find the prime factorization of each number: 12 = 2 × 2 × 3 and 18 = 2 × 3 × 3. The LCM is the product of all prime factors raised to the highest power: 2 × 2 × 3 × 3 = 36.

Problem 13: Convert the fraction 3/5 to a decimal.

Answer: 0.6

Explanation: Divide the numerator (3) by the denominator (5): 3 ÷ 5 = 0.6.

Problem 14: Convert 0.75 to a fraction in simplest form.

Answer: ¾

Explanation: 0.75 can be written as 75/100. Simplify the fraction by dividing both the numerator and denominator by their greatest common factor, which is 25: (75 ÷ 25) / (100 ÷ 25) = 3/4.

Problem 15: Calculate 25% of 80.

Answer: 20

Explanation: To find 25% of 80, multiply 80 by 0.25 (which is the decimal equivalent of 25%): 80 × 0.25 = 20. Alternatively, you can calculate ¼ of 80, which is also 20.

Section 4: Data Analysis and Probability (Problems 16-20)

Problem 16: Find the mean of the following data set: {5, 8, 12, 10, 7}

Answer: 8.4

Explanation: Add all the numbers together: 5 + 8 + 12 + 10 + 7 = 42. Then divide by the number of data points (5): 42 ÷ 5 = 8.4.

Problem 17: Find the median of the following data set: {3, 6, 1, 9, 4, 7}

Answer: 5.5

Explanation: First, arrange the data in ascending order: {1, 3, 4, 6, 7, 9}. Since there's an even number of data points, the median is the average of the middle two numbers: (4 + 6) / 2 = 5.

Problem 18: Find the mode of the following data set: {2, 5, 5, 8, 11, 5, 13}

Answer: 5

Explanation: The mode is the number that appears most frequently in the data set. In this case, 5 appears three times.

Problem 19: A bag contains 3 red marbles and 7 blue marbles. What is the probability of drawing a red marble?

Answer: 3/10

Explanation: The probability is the ratio of the number of favorable outcomes (drawing a red marble) to the total number of possible outcomes (total number of marbles): 3 / (3 + 7) = 3/10.

Problem 20: A coin is flipped twice. What is the probability of getting two heads?

Answer: ¼

Explanation: The possible outcomes when flipping a coin twice are: HH, HT, TH, TT. Only one outcome (HH) results in two heads. Therefore, the probability is 1/4.

Frequently Asked Questions (FAQ)

Q: What should I do if I get a different answer than the ones provided here?

A: Carefully review your steps. Double-check your calculations and ensure you're applying the correct formulas and methods. If you're still stuck, try working through the problem again step-by-step, or consider seeking help from a teacher or tutor. Identifying where you made a mistake is crucial for learning.

Q: Are these problems representative of what I might see on a test?

A: These problems cover fundamental concepts commonly tested in middle school and early high school mathematics and science. While specific problems might vary, understanding the underlying principles demonstrated here will equip you to tackle a wide range of similar questions.

Q: What resources can I use to further improve my understanding?

A: Explore online educational resources, practice worksheets, and educational videos that focus on the specific topics covered in this practice set. Many free and paid resources are available. Consistent practice is key to mastering these concepts.

Conclusion

This comprehensive answer key and explanation guide should provide you with a solid understanding of the fundamental concepts covered in this hypothetical "6-1 Additional Practice" assignment. Remember, mathematics and science build upon foundational knowledge; mastering these basics is essential for success in more advanced topics. Don't hesitate to review these explanations multiple times, practice similar problems, and seek help when needed. With consistent effort and a focused approach, you can achieve a deeper understanding and build confidence in your problem-solving abilities. Good luck with your studies!